This paper presents a way of studying the rationality that mathematics teachers utilize in managing the teaching of theorems in high school geometry. More generally, the study illustrates how to elicit the rationality that guides teachers in handling the demands of teaching practice. In particular, it illustrates how problematic classroom scenarios represented through animations of cartoon characters can facilitate thought experiments among groups of practitioners. Relying on video records from four study group sessions with experienced teachers of geometry, the study shows how these records can be parsed and inspected to identify categories of perception and appreciation with which experienced practitioners relate to an instance of an instructional situation. The study provides initial evidence that supports a theoretically-derived hypothesis, namely that teachers of geometry as a group recognize as normative the expectation that a teacher will sanction or endorse those propositions that are to be remembered as theorems for later use. In interacting with a story in which students had proven a proposition that the teacher had not identified as a theorem, the study also shows the kind of tactical resources that teachers of geometry could use to make it feasible for students to reuse such a proposition.The research reported in this article is supported by NSF, grant ESI-0353285 to Herbst and Chazan. Opinions expressed here are the sole responsibility of the authors and do not reflect the views of the Foundation.http://deepblue.lib.umich.edu/bitstream/2027.42/78023/1/ITH-PHTN&DC.pd
